Causality in Time Series - ClopiNet

Causal Search in SVARIf it is the case that the VAR process is unstable with the presence of cointegratingrelationships, it is more appropriate (Lütkepohl, 2006; Johansen, 2006) to estimate thefollowing re-parametrization of the VAR model, called Vector Error Correction Model(VECM):∆Y t = F 1 ∆Y t−1 + ... + F p−1 ∆Y t−p+1 − GY t−p + u t , (12)where F i = −(I k −A 1 −...−A i ), for i = 1,..., p−1 and G = I k −A 1 −...−A p . The (k×k)matrix G has rank r and thus G can be written as HC with H and C ′ of dimension (k×r)and of rank r. C ≡ [c 1 ,...,c r ] ′ is called the cointegrating matrix.Let ˜C, ˜H, and ˜F i be the maximum likelihood estimator of C, H, F according toJohansen’s (1988, 1991) approach. Then the asymptotic distribution of ˜Σ u , that is themaximum likelihood estimator of the covariance matrix of u t , is:√ dT [vech( ˜Σ u ) − vech(Σ u )] −→ N(0, 2D + k (Σ u ⊗ Σ u )D +′ ), (13)which is equivalent to equation (11) (see it again for the definition of the various operators).Thus, it turns out that the asymptotic distribution of the maximum likelihoodestimator ˜Σ u is the same as the OLS estimation ˆΣ u for the case of stable VAR.Thus, the application of the method described for testing residuals zero partial correlationscan be applied straightforwardly to cointegrated data. The model is estimatedas a VECM error correction model using Johansen’s (1988, 1991) approach, correlationsare tested exploiting the asymptotic distribution of ˜Σ u and finally can be parameterizedback in its VAR form of equation (3).2.4. Summary of the search procedureThe graphical causal models approach to SVAR identification, which we suggest incase of Gaussian and linear processes, can be summarized as follows.Step 1 Estimate the VAR model Y t = A 1 Y t−1 +...+A p Y t−p +u t with the usual specificationtests about normality, zero autocorrelation of residuals, lags, and unit roots (seeLütkepohl, 2006). If the hypothesis of nonstationarity is rejected, estimate the VARmodel via OLS (equivalent to MLE under the assumption of normality of the errors).If unit root tests do not reject I(1) nonstationarity in the data, specify the model asVECM testing the presence of cointegrating relationships. If tests suggest the presenceof cointegrating relationships, estimate the model as VECM. If cointegration is rejectedestimate the VAR models taking first difference.Step 2 Run tests for zero partial correlations between the elements u 1t ,...,u kt of u tusing the Wald statistics on the basis of the asymptotic distribution of the covariancematrix of u t . Note that not all possible partial correlations ρ(u it ,u jt |u ht ,...) need to betested, but only those necessary for step 3.k113